Odd index precomputation for authentication path computation

ABSTRACT

In one example an apparatus comprises a computer-readable memory, signature logic to compute a message hash of an input message using a secure hash algorithm, process the message hash to generate an array of secret key components for the input message, apply a hash chain function to the array of secret key components to generate an array of signature components, the hash chain function comprising a series of even-index hash chains and a series of odd-index hash chains, wherein the even-index hash chains and the odd-index hash chains generate a plurality of intermediate node values and a one-time public key component between the secret key components and the signature components and store at least some of the intermediate node values in the computer-readable memory for use in one or more subsequent signature operations. Other examples may be described.

BACKGROUND

Subject matter described herein relates generally to the field ofcomputer security and more particularly to odd index precomputation forauthentication path computation.

Existing public-key digital signature algorithms such asRivest-Shamir-Adleman (RSA) and Elliptic Curve Digital SignatureAlgorithm (ECDSA) are anticipated not to be secure against brute-forceattacks based on algorithms such as Shor's algorithm using quantumcomputers. As a result, there are efforts underway in the cryptographyresearch community and in various standards bodies to define newstandards for algorithms that are secure against quantum computers.

Accordingly, techniques to accelerate post-quantum signature schemessuch may find utility, e.g., in computer-based communication systems andmethods.

BRIEF DESCRIPTION OF THE DRAWINGS

The detailed description is described with reference to the accompanyingfigures.

FIGS. 1A and 1B are schematic illustrations of a one-time hash-basedsignatures scheme and a multi-time hash-based signatures scheme,respectively.

FIGS. 2A-2B are schematic illustrations of a one-time signature schemeand a multi-time signature scheme, respectively.

FIG. 3 is a schematic illustration of a signing device and a verifyingdevice, in accordance with some examples.

FIG. 4A is a schematic illustration of a Merkle tree structure, inaccordance with some examples.

FIG. 4B is a schematic illustration of a Merkle tree structure, inaccordance with some examples.

FIG. 5 is a schematic illustration of a compute blocks in anarchitecture to implement a signature algorithm, in accordance with someexamples.

FIG. 6A is a schematic illustration of a compute blocks in anarchitecture to implement signature generation in a signature algorithm,in accordance with some examples.

FIG. 6B is a schematic illustration of a compute blocks in anarchitecture to implement signature verification in a verificationalgorithm, in accordance with some examples.

FIG. 7 is a schematic illustration of a signature and verificationscheme, in accordance with some examples.

FIG. 8 is a schematic illustration of a signature and verificationscheme, in accordance with some examples.

FIG. 9 is a schematic illustration of a signature and verificationscheme, in accordance with some examples.

FIG. 10 is a schematic illustration of a signature and verificationscheme, in accordance with some examples.

FIG. 11 is a schematic illustration of a computing architecture whichmay be adapted to implement hardware acceleration in accordance withsome examples.

DETAILED DESCRIPTION

Described herein are exemplary systems and methods to implement oddindex precomputation for authentication path computation forpost-quantum cryptography secure hash-based signature algorithms. In thefollowing description, numerous specific details are set forth toprovide a thorough understanding of various examples. However, it willbe understood by those skilled in the art that the various examples maybe practiced without the specific details. In other instances,well-known methods, procedures, components, and circuits have not beenillustrated or described in detail so as not to obscure the examples.

As described briefly above, existing public-key digital signaturealgorithms such as Rivest-Shamir-Adleman (RSA) and Elliptic CurveDigital Signature Algorithm (ECDSA) are anticipated not to be secureagainst brute-force attacks based on algorithms such as Shor's algorithmusing quantum computers. The eXtended Merkle signature scheme (XMSS)and/or an eXtended Merkle many time signature scheme (XMSS-MT) arehash-based signature schemes that can protect against attacks by quantumcomputers. As used herein, the term XMSS shall refer to both the XMSSscheme and the XMSS-MT scheme.

An XMSS signature process implements a hash-based signature scheme usinga one-time signature scheme such as a Winternitz one-time signature(WOTS) or a derivative there of (e.g., WOTS+) in combination with asecure hash algorithm (SHA) such as SHA2-256 as the primary underlyinghash function. In some examples the XMSS signature/verification schememay also use one or more of SHA2-512, SHA3-SHAKE-256 or SHA3-SHAKE-512as secure hash functions. XMSS-specific hash functions include aPseudo-Random Function (PRF), a chain hash (F), a tree hash (H) andmessage hash function (H_(msg)). As used herein, the term WOTS shallrefer to the WOTS signature scheme and or a derivative scheme such asWOTS+.

The Leighton/Micali signature (LMS) scheme is another hash-basedsignature scheme that uses Leighton/Micali one-time signatures (LM-OTS)as the one-time signature building block. LMS signatures are based on aSHA2-256 hash function.

An XMSS signature process comprises three major operations. The firstmajor operation receives an input message (M) and a private key (sk) andutilizes a one-time signature algorithm (e.g., WOTS+) to generate amessage representative (M′) that encodes a public key (pk). In a 128-bitpost quantum security implementation the input message M is subjected toa hash function and then divided into 67 message components (n byteseach), each of which are subjected to a hash chain function to generatethe corresponding 67 components of the digital signature. Each chainfunction invokes a series of underlying secure hash algorithms (SHA).

The second major operation is an L-Tree computation, which combinesWOTS+ (or WOTS) public key components (n-bytes each) and produces asingle n-byte value. For example, in the 128-bit post-quantum securitythere are 67 public key components, each of which invokes an underlyingsecure hash algorithm (SHA) that is performed on an input block.

The third major operation is a tree-hash operation, which constructs aMerkle tree. In an XMSS verification, an authentication path that isprovided as part of the signature and the output of L-tree operation isprocessed by a tree-hash operation to generate the root node of theMerkle tree, which should correspond to the XMSS public key. For XMSSverification with 128-bit post-quantum security, traversing the Merkletree comprises executing secure hash operations. In an XMSSverification, the output of the Tree-hash operation is compared with theknown public key. If they match, then the signature is accepted. Bycontrast, if they do not match then the signature is rejected.

The XMSS signature process is computationally expensive. An XMSSsignature process invokes hundreds, or even thousands, of cycles of hashcomputations. Subject matter described herein addresses these and otherissues by providing systems and methods to implement accelerators forpost-quantum cryptography secure XMSS and LMS hash-based signing andverification.

Post-Quantum Cryptography Overview

Post-Quantum Cryptography (also referred to as “quantum-proof”,“quantum-safe”, “quantum-resistant”, or simply “PQC”) takes a futuristicand realistic approach to cryptography. It prepares those responsiblefor cryptography as well as end-users to know the cryptography isoutdated; rather, it needs to evolve to be able to successfully addressthe evolving computing devices into quantum computing and post-quantumcomputing.

It is well-understood that cryptography allows for protection of datathat is communicated online between individuals and entities and storedusing various networks. This communication of data can range fromsending and receiving of emails, purchasing of goods or services online,accessing banking or other personal information using websites, etc.

Conventional cryptography and its typical factoring and calculating ofdifficult mathematical scenarios may not matter when dealing withquantum computing. These mathematical problems, such as discretelogarithm, integer factorization, and elliptic-curve discrete logarithm,etc., are not capable of withstanding an attack from a powerful quantumcomputer. Although any post-quantum cryptography could be built on thecurrent cryptography, the novel approach would need to be intelligent,fast, and precise enough to resist and defeat any attacks by quantumcomputers

Today's PQC is mostly focused on the following approaches: 1) hash-basedcryptography based on Merkle's hash tree public-key signature system of1979, which is built upon a one-message-signature idea of Lamport andDiffie; 2) code-based cryptography, such as McEliece's hidden-Goppa-codepublic-key encryption system; 3) lattice-based cryptography based onHoffstein-Pipher-Silverman public-key-encryption system of 1998; 4)multivariate-quadratic equations cryptography based on Patarin's HFEpublic-key-signature system of 1996 that is further based on theMatumoto-Imai proposal; 5) supersingular elliptical curve isogenycryptography that relies on supersingular elliptic curves andsupersingular isogeny graphs; and 6) symmetric key quantum resistance.

FIGS. 1A and 1B illustrate a one-time hash-based signatures scheme and amulti-time hash-based signatures scheme, respectively. As aforesaid,hash-based cryptography is based on cryptographic systems like Lamportsignatures, Merkle Signatures, extended Merkle signature scheme (XMSS),and SPHINCs scheme, etc. With the advent of quantum computing and inanticipation of its growth, there have been concerns about variouschallenges that quantum computing could pose and what could be done tocounter such challenges using the area of cryptography.

One area that is being explored to counter quantum computing challengesis hash-based signatures (HBS) since these schemes have been around fora long while and possess the necessarily basic ingredients to counterthe quantum counting and post-quantum computing challenges. HBS schemesare regarded as fast signature algorithms working with fast platformsecured-boot, which is regarded as the most resistant to quantum andpost-quantum computing attacks.

For example, as illustrated with respect to FIG. 1A, a scheme of HBS isshown that uses Merkle trees along with a one-time signature (OTS)scheme 100, such as using a private key to sign a message and acorresponding public key to verify the OTS message, where a private keyonly signs a single message.

Similarly, as illustrated with respect to FIG. 1B, another HBS scheme isshown, where this one relates to multi-time signatures (MTS) scheme 150,where a private key can sign multiple messages.

FIGS. 2A and 2B illustrate a one-time signature scheme and a multi-timesignature scheme, respectively. Continuing with HBS-based OTS scheme 100of FIG. 1A and MTS scheme 150 of FIG. 1B, FIG. 2A illustrates WinternitzOTS scheme 200, which was offered by Robert Winternitz of StanfordMathematics Department publishing as hw(x) as opposed to h(x)lh(y),while FIG. 2B illustrates XMSS MTS scheme 250, respectively.

For example, WOTS scheme 200 of FIG. 2A provides for hashing and parsingof messages into M, with 67 integers between [0, 1, 2, . . . , 15], suchas private key, sk, 205, signature, s, 210, and public key, pk, 215,with each having 67 components of 32 bytes each.

FIG. 2B illustrates XMSS MTS scheme 250 that allows for a combination ofWOTS scheme 200 of FIG. 2A and XMSS scheme 255 having XMSS Merkle tree.As discussed previously with respect to FIG. 2A, WOTs scheme 200 isbased on a one-time public key, pk, 215, having 67 components of 32bytes each, that is then put through L-Tree compression algorithm 260 tooffer WOTS compressed pk 265 to take a place in the XMSS Merkle tree ofXMSS scheme 255. It is contemplated that XMSS signature verification mayinclude computing WOTS verification and checking to determine whether areconstructed root node matches the XMSS public key, such as rootnode=XMSS public key.

Post-Quantum Cryptography Algorithms

FIG. 3 is a schematic illustration of a high-level architecture of asecure environment 300 that includes a first device 310 and a seconddevice 350, in accordance with some examples. Referring to FIG. 3, eachof the first device 310 and the second device 350 may be embodied as anytype of computing device capable of performing the functions describedherein. For example, in some embodiments, each of the first device 310and the second device 350 may be embodied as a laptop computer, tabletcomputer, notebook, netbook, Ultrabook™, a smartphone, cellular phone,wearable computing device, personal digital assistant, mobile Internetdevice, desktop computer, router, server, workstation, and/or any othercomputing/communication device.

First device 310 includes one or more processor(s) 320 and a memory 322to store a private key 324. The processor(s) 320 may be embodied as anytype of processor capable of performing the functions described herein.For example, the processor(s) 320 may be embodied as a single ormulti-core processor(s), digital signal processor, microcontroller, orother processor or processing/controlling circuit Similarly, the memory322 may be embodied as any type of volatile or non-volatile memory ordata storage capable of performing the functions described herein. Inoperation, the memory 322 may store various data and software usedduring operation of the first device 310 such as operating systems,applications, programs, libraries, and drivers. The memory 322 iscommunicatively coupled to the processor(s) 320. In some examples theprivate key 324 may reside in a secure memory that may be part memory322 or may be separate from memory 322.

First device 310 further comprises authentication logic 330 whichincludes memory 332, signature logic, and verification logic 336. Hashlogic 332 is configured to hash (i.e., to apply a hash function to) amessage (M) to generate a hash value (m′) of the message M. Hashfunctions may include, but are not limited to, a secure hash function,e.g., secure hash algorithms SHA2-256 and/or SHA3-256, etc. SHA2-256 maycomply and/or be compatible with Federal Information ProcessingStandards (FIPS) Publication 180-4, titled: “Secure Hash Standard(SHS)”, published by National Institute of Standards and Technology(NIST) in March 2012, and/or later and/or related versions of thisstandard. SHA3-256 may comply and/or be compatible with FIPS Publication202, titled: “SHA-3 Standard: Permutation-Based Hash andExtendable-Output Functions”, published by NIST in August 2015, and/orlater and/or related versions of this standard.

Signature logic 332 may be configured to generate a signature to betransmitted, i.e., a transmitted signature. In instances in which thefirst device 310 is the signing device, the transmitted signature mayinclude a number, L, of transmitted signature elements with eachtransmitted signature element corresponding to a respective messageelement. For example, for each message element, m_(i), signature logic332 may be configured to perform a selected signature operation on eachprivate key element, sk_(i) of the private key, sk, a respective numberof times related to a value of each message element, m_(i) included inthe message representative m′. For example, signature logic 332 may beconfigured to apply a selected hash function to a corresponding privatekey element, sk_(i), m_(i) times. In another example, signature logic332 may be configured to apply a selected chain function (that containsa hash function) to a corresponding private key element, sk_(i), m_(i)times. The selected signature operations may, thus, correspond to aselected hash-based signature scheme.

As described above, hash-based signature schemes may include, but arenot limited to, a Winternitz (W) one time signature (OTS) scheme, anenhanced Winternitz OTS scheme (e.g., WOTS+), a Merkle many timesignature scheme, an extended Merkle signature scheme (XMSS) and/or anextended Merkle multiple tree signature scheme (XMSS-MT), etc. Hashfunctions may include, but are not limited to SHA2-256 and/or SHA3-256,etc. For example, XMSS and/or XMSS-MT may comply or be compatible withone or more Internet Engineering Task Force (IETF®) informational draftInternet notes, e.g., “XMSS: Extended Hash-Based Signatures, releasedMay, 2018, by the Internet Research Task Force (IRTF), Crypto ForumResearch Group which may be found athttps://tools.ietf.org/html/rfc8391.

A WOTS signature algorithm may be used to generate a signature and toverify a received signature utilizing a hash function. WOTS is furtherconfigured to use the private key and, thus, each private key element,sk_(i), one time. For example, WOTS may be configured to apply a hashfunction to each private key element, m_(i), or N-m_(i) times togenerate a signature and to apply the hash function to each receivedmessage element N-m_(i′) or m_(i′) times to generate a correspondingverification signature element. The Merkle many time signature scheme isa hash-based signature scheme that utilizes an OTS and may use a publickey more than one time. For example, the Merkle signature scheme mayutilize Winternitz OTS as the one-time signature scheme. WOTS+ isconfigured to utilize a family of hash functions and a chain function.

XMSS, WOTS+ and XMSS-MT are examples of hash-based signature schemesthat utilize chain functions. Each chain function is configured toencapsulate a number of calls to a hash function and may further performadditional operations. In some examples, the number of calls to the hashfunction included in the chain function may be fixed. Chain functionsmay improve security of an associated hash-based signature scheme.

Cryptography logic 340 is configured to perform various cryptographicand/or security functions on behalf of the signing device 310. In someembodiments, the cryptography logic 340 may be embodied as acryptographic engine, an independent security co-processor of thesigning device 310, a cryptographic accelerator incorporated into theprocessor(s) 320, or a standalone software/firmware. In someembodiments, the cryptography logic 340 may generate and/or utilizevarious cryptographic keys (e.g., symmetric/asymmetric cryptographickeys) to facilitate encryption, decryption, signing, and/or signatureverification. Additionally, in some embodiments, the cryptography logic340 may facilitate to establish a secure connection with remote devicesover communication link. It should further be appreciated that, in someembodiments, the cryptography module 340 and/or another module of thefirst device 310 may establish a trusted execution environment or secureenclave within which a portion of the data described herein may bestored and/or a number of the functions described herein may beperformed.

After the signature is generated as described above, the message, M, andsignature may then be sent by first device 310, e.g., via communicationlogic 342, to second device 350 via network communication link 390. Inan embodiment, the message, M, may not be encrypted prior totransmission. In another embodiment, the message, M, may be encryptedprior to transmission. For example, the message, M, may be encrypted bycryptography logic 340 to produce an encrypted message.

Second device 350 may also include one or more processors 360 and amemory 362 to store a public key 364. As described above, theprocessor(s) 360 may be embodied as any type of processor capable ofperforming the functions described herein. For example, the processor(s)360 may be embodied as a single or multi-core processor(s), digitalsignal processor, microcontroller, or other processor orprocessing/controlling circuit. Similarly, the memory 362 may beembodied as any type of volatile or non-volatile memory or data storagecapable of performing the functions described herein. In operation, thememory 362 may store various data and software used during operation ofthe second device 350 such as operating systems, applications, programs,libraries, and drivers. The memory 362 is communicatively coupled to theprocessor(s) 360.

In some examples the public key 364 may be provided to second device 350in a previous exchange. The public key, p_(k), is configured to containa number L of public key elements, i.e., p_(k)=[pk₁, . . . , pk_(L)].The public key 364 may be stored, for example, to memory 362.

Second device 350 further comprises authentication logic 370 whichincludes hash logic 372, signature logic, and verification logic 376. Asdescribed above, hash logic 372 is configured to hash (i.e., to apply ahash function to) a message (M) to generate a hash message (m′). Hashfunctions may include, but are not limited to, a secure hash function,e.g., secure hash algorithms SHA2-256 and/or SHA3-256, etc. SHA2-256 maycomply and/or be compatible with Federal Information ProcessingStandards (FIPS) Publication 180-4, titled: “Secure Hash Standard(SHS)”, published by National Institute of Standards and Technology(NIST) in March 2012, and/or later and/or related versions of thisstandard. SHA3-256 may comply and/or be compatible with FIB Publication202, titled: “SHA-3 Standard: Permutation-Based Hash andExtendable-Output Functions”, published by NIST in August 2015, and/orlater and/or related versions of this standard.

In instances in which the second device is the verifying device,authentication logic 370 is configured to generate a verificationsignature based, at least in part, on the signature received from thefirst device and based, at least in part, on the received messagerepresentative (m′). For example, authentication logic 370 mayconfigured to perform the same signature operations, i.e., apply thesame hash function or chain function as applied by hash logic 332 ofauthentication logic 330, to each received message element a number,N-m_(i′) (or m_(i′)), times to yield a verification message element.Whether a verification signature, i.e., each of the L verificationmessage elements, corresponds to a corresponding public key element,pk_(i), may then be determined. For example, verification logic 370 maybe configured to compare each verification message element to thecorresponding public key element, p_(ki). If each of the verificationmessage element matches the corresponding public key element, p_(ki),then the verification corresponds to success. In other words, if all ofthe verification message elements match the public key elements, p_(k1),. . . , pk_(L), then the verification corresponds to success. If anyverification message element does not match the corresponding public keyelement, pk_(i), then the verification corresponds to failure.

As described in greater detail below, in some examples theauthentication logic 330 of the first device 310 includes one or moreaccelerators 338 that cooperate with the hash logic 332, signature logic334 and/or verification logic 336 to accelerate authenticationoperations. Similarly, in some examples the authentication logic 370 ofthe second device 310 includes one or more accelerators 378 thatcooperate with the hash logic 372, signature logic 374 and/orverification logic 376 to accelerate authentication operations. Examplesof accelerators are described in the following paragraphs and withreference to the accompanying drawings.

The various modules of the environment 300 may be embodied as hardware,software, firmware, or a combination thereof. For example, the variousmodules, logic, and other components of the environment 300 may form aportion of, or otherwise be established by, the processor(s) 320 offirst device 310 or processor(s) 360 of second device 350, or otherhardware components of the devices As such, in some embodiments, one ormore of the modules of the environment 300 may be embodied as circuitryor collection of electrical devices (e.g., an authentication circuitry,a cryptography circuitry, a communication circuitry, a signaturecircuitry, and/or a verification circuitry). Additionally, in someembodiments, one or more of the illustrative modules may form a portionof another module and/or one or more of the illustrative modules may beindependent of one another.

FIG. 4A is a schematic illustration of a Merkle tree structureillustrating signing operations, in accordance with some examples.Referring to FIG. 4A, an XMSS signing operation requires theconstruction of a Merkle tree 400A using the local public key from eachleaf WOTS node 410 to generate a global public key (PK) 420. In someexamples the authentication path and the root node value can be computedoff-line such that these operations do not limit performance. Each WOTSnode 410 has a unique secret key, “sk” which is used to sign a messageonly once. The XMSS signature consists of a signature generated for theinput message and an authentication path of intermediate tree nodes toconstruct the root of the Merkle tree.

FIG. 4B is a schematic illustration of a Merkle tree structure 400Bduring verification, in accordance with some examples. Duringverification, the input message and signature are used to compute thelocal public key 420B of the WOTS node, which is further used to computethe tree root value using the authentication path. A successfulverification will match the computed tree root value to the public keyPK shared by the signing entity. The WOTS and L-Tree operationsconstitute a significant portion of XMSS sign/verify latencyrespectively, thus defining the overall performance of theauthentication system. Described herein are various pre-computationtechniques which may be implemented to speed-up authentication pathcomputations, thereby improving XMSS performance. The techniques areapplicable to the other hash options and scale well for both softwareand hardware implementations.

FIG. 5 is a schematic illustration of a compute blocks in anarchitecture 500 to implement a signature algorithm, in accordance withsome examples. Referring to FIG. 5, the WOTS+ operation involves 67parallel chains of 16 SHA2-256 HASH functions, each with the secret keysk[66:0] as input. Each HASH operation in the chain consists of 2pseudo-random functions (PRF) using SHA2-256 to generate a bitmask and akey. The bitmask is XOR-ed with the previous hash and concatenated withthe key as input message to a 3rd SHA2-256 hash operation. The67×32-byte WOTS public key pk[66:0] is generated by hashing secret keysk across the 67 hash chains.

FIG. 6A is a schematic illustration of a compute blocks in anarchitecture 600A to implement signature generation in a signaturealgorithm, in accordance with some examples. As illustrated in FIG. 6A,for message signing, the input message is hashed and pre-processed tocompute a 67×4-bit value, which is used as an index to choose anintermediate hash value in each operation of the chain function.

FIG. 6B is a schematic illustration of a compute blocks in anarchitecture 600B to implement signature verification in a verificationalgorithm, in accordance with some examples. Referring to FIG. 6B,during verification, the message is again hashed to compute thesignature indices and compute the remaining HASH operations in eachchain to compute the WOTS public key pk. This value and theauthentication path are used to compute the root of the Merkle tree andcompare with the shared public key PK to verify the message.

Odd Index Precomputation for Authentication Path Computation

As described above, Hash-Based Signature (HBS) algorithms such as XMSSand LMS schemes offer a promising approach for post-quantum digitalsignatures. HBS algorithms use one-time signature algorithms as afundamental building block. The main limitation of one-time signaturealgorithms is that each key pair can sign only a single message. HBSschemes such as XMSS and LMS may use processes such as an L-Treecomputation to combine a large number of one-time key pairs into asingle multi-time HBS key pair. To generate a signature of a givenmessage, the signing device signs the message using one unused one-timekey pair to produce a one-time signature of the message. Additionally,the signing device needs to produce an Authentication Path of thatsignature through a Merkle tree structure. This authentication path isused to confirm that the used one-time key pair is part of the set ofone-time key pairs represented by the single multi-time HBS key pair.For signature verification, the verifying device at first checks thatthe one-time signature is authentic and then by using the authenticationpath the verifier checks that the signature was produced by a validone-time key pair.

FIG. 7 is a schematic illustration of a one-time signature andverification scheme, in accordance with some examples. Referring to FIG.7, in some examples the one-time signature keygen/sign/verify algorithms700 operate on a message which is spread over 67 chunks of 32 byteseach. More precisely, the private key (sk) is composed of 67 chunks of32 bytes each (sk₁ . . . sk₆₇), the signature (s) is composed by 67chunks of 32 bytes each (s₁ . . . s₆₇), and the public key (pk) iscomposed by 67 chunks of 32 bytes each (pk₁ . . . pk₆₇). To generate thepublic key (pk) from the private key (sk), the one-time algorithmapplies a hash chain function 15 times. The signature (s) of a message mis generated as follows. At first, the message is hashed and thenencoded into 67 integers between 0 and 15. The signature of the messagem is the result of applying the hash chain over the private key chunksk_(i) exactly m_(i) times, where m_(i) denotes the i-th integer thatrepresents (in encoded format) the message to be signed.

Thus, given a private key (pk) and a message (m) it is possible tocompute the signature (s). Given the signature (s) and the message (m),it is possible to reconstruct the associated one-time public key (pk) atthe verifying device without access to the private key (sk). If thereconstructed public key (pk) matches the one-time public key (pk) ofthe signing device, then the signature is deemed authentic. If not, itis deemed rejected as not authentic.

HBS signatures such as XMSS are composed by one-time signatures plus aset of 32-bytes nodes referred to as an authentication path. Forsignatures of even index, the first node of the authentication path is acompressed version of the one-time public key (pk) of the subsequent oddindex that is used to verify the next message. Also, to compute aone-time public key (pk), the user needs to compute 67×15 different32-bytes intermediate values. Among these 67×15 intermediate values,there are 67 of them (in a specific order) that convey the next one-timesignature (i.e., the message to be signed will select which value and atwhich order such values are picked). Thus, the 67×15 intermediate valuesrequired in the authentication path computation of signatures of evenindex may be stored to save the cost of generating signatures of oddindex. In some examples a similar approach can be used to partially savethe cost of generating authentication path of signatures of odd index.

FIG. 8 is a schematic illustration of a WOTS signature and verificationscheme 800, in accordance with some examples. Referring to FIG. 8, HBSschemes combine a large number of one-time key pairs (pk₁, pk₂, . . . ,pk₆₇) into one single multi-time key pair. This may be performed byusing a Merkle tree which, as described above, is a binary tree that hasleaf nodes that are compressed versions of the one-time public keys.This compression of the one-time public key (pk) is done by an L-TreeCompression algorithm 820 which compresses 67 chunks of 32 bytes into acompressed public key 822 of 32 bytes, which is input as the first nodeof the Merkle tree. As described above, given the leaf nodes, the Merkletree is built by generating a parent node as the hash of the twochildren nodes concatenated. Therefore, it is possible to construct theroot node of a Merkle tree given its leaf nodes. The Merkle root nodecorresponds to the HBS multi-time public-key.

FIG. 8 illustrates how a one-time public key is plugged into a Merkletree. Given a one-time public key (pk) and the intermediate nodes 830,832, 834, 836 filled in a light-shaded gray (i.e., authentication pathnodes) a verifier can reconstruct the root node of the Merkle tree byfollowing the construction rule mentioned above. The verifier checks ifthe signature is authentic by reconstructing the root node and verifyingthat the root node matches with the HBS public key (then signature isrejected otherwise).

FIG. 9 is a schematic illustration of a signature and verificationscheme 900, in accordance with some examples. Referring to FIG. 9, it isnoted that the first node of the authentication path of a signature ofeven index (for example, 0, 2, 4, . . . ) consists of the compressedversion of the one-time public key (pk) related to the next (odd)signature (for example, 1, 3, 5, . . . ). The one-time public key (pk)is constructed from the associated one-time private key (sk), asdescribed above. As illustrated in FIG. 7, inbetween the one-timeprivate key (sk) and the one-time public key (pk), there areintermediate nodes that will compose the signature (s). Therefore, insome examples a signing device can store in a computer-readable memoryall such intermediate nodes between the one-time private key (sk) andthe one-time public key (pk) which are used to generate the firstauthentication path node. FIG. 8 highlights one example of theintermediate nodes 830, 832, 834, 836 that need to be stored during thisprocess. In some examples, this technique requires a total of (67×15×32bytes=≈32 KB) of additional memory to be allocated during the signatureprocess, However, this memory overhead saves the cost of generatingsignatures having odd index. In scenarios in which the additional memoryoverhead is a limitation, selective intermediate nodes can be storedduring authentication path computation and used as start point duringactual message signing. This provides a trade-off between memoryoverhead and increasing the speed of the signature process signing.

FIG. 10 is a schematic illustration of a signature and verificationscheme 1000, in accordance with some examples. Referring to FIG. 10, insome examples a similar approach may be used to improve the efficiencyof authentication path computation of odd index components. During thesignature generation procedure for an even-index such a procedure, thesigning device necessarily computes intermediate nodes in hash chainsthat, if completed, would result in a one-time public key that will berequired as the first node of the authentication path of the next(odd-index) signature. Thus, in some examples the signing device maycontinue computing hash chains from the signature to the point at whichthe associated one-time public key (pk) is obtained. These public keycomponents may then be provided with the signature to the verifyingdevice. In other examples the verifying device may receive the signatureand, during verification, compute the one-time public key and store itin memory. The authentication path node related to this public key doesnot need to be transmitted by the signing device since the signingdevice already has it from previous signature verification. Thus, boththe signing device and the verifying device can do utilize thistechnique of computing the one-time public key needed for next signatureauthentication path.

This pre-computation would require 67×(15/2) additional hash chaincomputations during the signature process to compute the one-time publickey, on average, as opposed to 67×15 hash chain computations required ina conventional implementation that would build such one-time public keyfrom the one-time private key, instead of from the signature. Therefore,this precomputation offers a 50% speedup in the process of building thefirst node of the authentication of odd-index signatures, on average.

FIG. 11 illustrates an embodiment of an exemplary computing architecturethat may be suitable for implementing various embodiments as previouslydescribed. In various embodiments, the computing architecture 1100 maycomprise or be implemented as part of an electronic device. In someembodiments, the computing architecture 1100 may be representative, forexample of a computer system that implements one or more components ofthe operating environments described above. In some embodiments,computing architecture 1100 may be representative of one or moreportions or components of a DNN training system that implement one ormore techniques described herein. The embodiments are not limited inthis context.

As used in this application, the terms “system” and “component” and“module” are intended to refer to a computer-related entity, eitherhardware, a combination of hardware and software, software, or softwarein execution, examples of which are provided by the exemplary computingarchitecture 1100. For example, a component can be, but is not limitedto being, a process running on a processor, a processor, a hard diskdrive, multiple storage drives (of optical and/or magnetic storagemedium), an object, an executable, a thread of execution, a program,and/or a computer. By way of illustration, both an application runningon a server and the server can be a component. One or more componentscan reside within a process and/or thread of execution, and a componentcan be localized on one computer and/or distributed between two or morecomputers. Further, components may be communicatively coupled to eachother by various types of communications media to coordinate operations.The coordination may involve the uni-directional or bi-directionalexchange of information. For instance, the components may communicateinformation in the form of signals communicated over the communicationsmedia. The information can be implemented as signals allocated tovarious signal lines. In such allocations, each message is a signal.Further embodiments, however, may alternatively employ data messages.Such data messages may be sent across various connections. Exemplaryconnections include parallel interfaces, serial interfaces, and businterfaces.

The computing architecture 1100 includes various common computingelements, such as one or more processors, multi-core processors,co-processors, memory units, chipsets, controllers, peripherals,interfaces, oscillators, timing devices, video cards, audio cards,multimedia input/output (I/O) components, power supplies, and so forth.The embodiments, however, are not limited to implementation by thecomputing architecture 1100.

As shown in FIG. 11, the computing architecture 1100 includes one ormore processors 1102 and one or more graphics processors 1108, and maybe a single processor desktop system, a multiprocessor workstationsystem, or a server system having a large number of processors 1102 orprocessor cores 1107. In on embodiment, the system 1100 is a processingplatform incorporated within a system-on-a-chip (SoC or SOC) integratedcircuit for use in mobile, handheld, or embedded devices.

An embodiment of system 1100 can include, or be incorporated within aserver-based gaming platform, a game console, including a game and mediaconsole, a mobile gaming console, a handheld game console, or an onlinegame console. In some embodiments system 1100 is a mobile phone, smartphone, tablet computing device or mobile Internet device. Dataprocessing system 1100 can also include, couple with, or be integratedwithin a wearable device, such as a smart watch wearable device, smarteyewear device, augmented reality device, or virtual reality device. Insome embodiments, data processing system 1100 is a television or set topbox device having one or more processors 1102 and a graphical interfacegenerated by one or more graphics processors 1108.

In some embodiments, the one or more processors 1102 each include one ormore processor cores 1107 to process instructions which, when executed,perform operations for system and user software. In some embodiments,each of the one or more processor cores 1107 is configured to process aspecific instruction set 1109. In some embodiments, instruction set 1109may facilitate Complex Instruction Set Computing (CISC), ReducedInstruction Set Computing (RISC), or computing via a Very LongInstruction Word (VLIW). Multiple processor cores 1107 may each processa different instruction set 1109, which may include instructions tofacilitate the emulation of other instruction sets. Processor core 1107may also include other processing devices, such a Digital SignalProcessor (DSP).

In some embodiments, the processor 1102 includes cache memory 1104.Depending on the architecture, the processor 1102 can have a singleinternal cache or multiple levels of internal cache. In someembodiments, the cache memory is shared among various components of theprocessor 1102. In some embodiments, the processor 1102 also uses anexternal cache (e.g., a Level-3 (L3) cache or Last Level Cache (LLC))(not shown), which may be shared among processor cores 1107 using knowncache coherency techniques. A register file 1106 is additionallyincluded in processor 1102 which may include different types ofregisters for storing different types of data (e.g., integer registers,floating point registers, status registers, and an instruction pointerregister). Some registers may be general-purpose registers, while otherregisters may be specific to the design of the processor 1102.

In some embodiments, one or more processor(s) 1102 are coupled with oneor more interface bus(es) 1110 to transmit communication signals such asaddress, data, or control signals between processor 1102 and othercomponents in the system. The interface bus 1110, in one embodiment, canbe a processor bus, such as a version of the Direct Media Interface(DMI) bus. However, processor busses are not limited to the DMI bus, andmay include one or more Peripheral Component Interconnect buses (e.g.,PCI, PCI Express), memory busses, or other types of interface busses. Inone embodiment the processor(s) 1102 include an integrated memorycontroller 1116 and a platform controller hub 1130. The memorycontroller 1116 facilitates communication between a memory device andother components of the system 1100, while the platform controller hub(PCH) 1130 provides connections to I/O devices via a local I/O bus.

Memory device 1120 can be a dynamic random-access memory (DRAM) device,a static random-access memory (SRAM) device, flash memory device,phase-change memory device, or some other memory device having suitableperformance to serve as process memory. In one embodiment the memorydevice 1120 can operate as system memory for the system 1100, to storedata 1122 and instructions 1121 for use when the one or more processors1102 executes an application or process. Memory controller hub 1116 alsocouples with an optional external graphics processor 1112, which maycommunicate with the one or more graphics processors 1108 in processors1102 to perform graphics and media operations. In some embodiments adisplay device 1111 can connect to the processor(s) 1102. The displaydevice 1111 can be one or more of an internal display device, as in amobile electronic device or a laptop device or an external displaydevice attached via a display interface (e.g., DisplayPort, etc.). Inone embodiment the display device 1111 can be a head mounted display(HMD) such as a stereoscopic display device for use in virtual reality(VR) applications or augmented reality (AR) applications.

In some embodiments the platform controller hub 1130 enables peripheralsto connect to memory device 1120 and processor 1102 via a high-speed I/Obus. The I/O peripherals include, but are not limited to, an audiocontroller 1146, a network controller 1134, a firmware interface 1128, awireless transceiver 1126, touch sensors 1125, a data storage device1124 (e.g., hard disk drive, flash memory, etc.). The data storagedevice 1124 can connect via a storage interface (e.g., SATA) or via aperipheral bus, such as a Peripheral Component Interconnect bus (e.g.,PCI, PCI Express). The touch sensors 1125 can include touch screensensors, pressure sensors, or fingerprint sensors. The wirelesstransceiver 1126 can be a Wi-Fi transceiver, a Bluetooth transceiver, ora mobile network transceiver such as a 3G, 4G, or Long Term Evolution(LTE) transceiver. The firmware interface 1128 enables communicationwith system firmware, and can be, for example, a unified extensiblefirmware interface (UEFI). The network controller 1134 can enable anetwork connection to a wired network. In some embodiments, ahigh-performance network controller (not shown) couples with theinterface bus 1110. The audio controller 1146, in one embodiment, is amulti-channel high definition audio controller. In one embodiment thesystem 1100 includes an optional legacy I/O controller 1140 for couplinglegacy (e.g., Personal System 2 (PS/2)) devices to the system. Theplatform controller hub 1130 can also connect to one or more UniversalSerial Bus (USB) controllers 1142 connect input devices, such askeyboard and mouse 1143 combinations, a camera 1144, or other USB inputdevices.

The following pertains to further examples.

Example 1 is an comprising a computer-readable memory; signature logicto compute a message hash of an input message using a secure hashalgorithm; process the message hash to generate an array of secret keycomponents for the input message; apply a hash chain function to thearray of secret key components to generate an array of signaturecomponents, the hash chain function comprising a series of even-indexhash chains and a series of odd-index has chains, wherein the even-indexhash chains and the odd-index hash chains generate a plurality ofintermediate node values and a one-time public key component between thesecret key components and the signature components; and store at leastsome of the intermediate node values in the computer-readable memory foruse in one or more subsequent signature operations.

In Example 2, the subject matter of Example 1 can optionally includelogic to compute the message hash of the input message using at leastone of a Winterniz One Time Signature (WOTS) scheme or a WOTS+ schemethat invokes a secure hash algorithm (SHA) hash function.

In Example 3, the subject matter of any one of Examples 1-2 canoptionally include logic to store the intermediate node values generatedand one time public key components generated by the even-index hashchains in the computer-readable memory; and use the one-time public keycomponents to define computations through an authentication path througha Merkle tree.

In Example 4, the subject matter of any one of Examples 1-3 canoptionally include logic to apply a L-tree operation to the public keycomponents to compress the public key components into a single leaf nodevalue; and provide the single leaf node value as an input to the Merkletree.

In Example 5, the subject matter of any one of Examples 1-4 canoptionally include logic to perform a first hash operations using thesingle leaf node value and a first odd-index public key component togenerate a first parent node in the Merkle tree; and perform a series ofhash operations using subsequent odd-index public key components todetermine a root node value of the Merkle tree.

In Example 6, the subject matter of any one of Examples 1-5 canoptionally include logic to compare the root node value of the Merkletree to a multi-signature public key value associated with the signingdevice; and generate an authentication signal when the root node valueof the Merkle tree matches the multi-signature public key valueassociated with the signing device.

In Example 7, the subject matter of any one of Examples 1-6 canoptionally include logic to compare the root node value of the Merkletree to a multi-signature public key value associated with the signingdevice; and generate an authentication signal when the root node valueof the Merkle tree matches the multi-signature public key valueassociated with the signing device.

Example 8 is a computer-implemented method, comprising computing amessage hash of an input message using a secure hash algorithm;processing the message hash to generate an array of secret keycomponents for the input message; applying a hash chain function to thearray of secret key components to generate an array of signaturecomponents, the hash chain function comprising a series of even-indexhash chains and a series of odd-index has chains, wherein the even-indexhash chains and the odd-index hash chains generate a plurality ofintermediate node values and a one-time public key component between thesecret key components and the signature components; and storing at leastsome of the intermediate node values in the computer-readable memory foruse in one or more subsequent signature operations.

In Example 9, the subject matter of Example 8 can optionally includecomputing the message hash of the input message using at least one of aWinterniz One Time Signature (WOTS) scheme or a WOTS+ scheme thatinvokes a secure hash algorithm (SHA) hash function.

In Example 10, the subject matter of any one of Examples 8-9 canoptionally include storing the intermediate node values generated andone time public key components generated by the even-index hash chainsin the computer-readable memory; and using the one-time public keycomponents to define computations through an authentication path througha Merkle tree.

In Example 11, the subject matter of any one of Examples 8-10 canoptionally include applying a L-tree operation to the public keycomponents to compress the public key components into a single leaf nodevalue; and providing the single leaf node value as an input to theMerkle tree.

In Example 12 the subject matter of any one of Examples 8-11 canoptionally include performing a first hash operations using the singleleaf node value and a first odd-index public key component to generate afirst parent node in the Merkle tree; and performing a series of hashoperations using subsequent odd-index public key components to determinea root node value of the Merkle tree.

In Example 13 the subject matter of any one of Examples 8-12 canoptionally include comparing the root node value of the Merkle tree to amulti-signature public key value associated with the signing device; andgenerating an authentication signal when the root node value of theMerkle tree matches the multi-signature public key value associated withthe signing device.

In Example 14 the subject matter of any one of Examples 8-13 canoptionally include comparing the root node value of the Merkle tree to amulti-signature public key value associated with the signing device; andgenerating an authentication fail signal when the root node value of theMerkle tree does not match the multi-signature public key valueassociated with the signing device.

Example 15 is a non-transitory computer-readable medium comprisinginstructions which, when executed by a processor, configure theprocessor to perform operations, comprising storing a public keyassociated with a signing device in a computer-readable medium;computing a message hash of an input message using a secure hashalgorithm; processing the message hash to generate an array of secretkey components for the input message; applying a hash chain function tothe array of secret key components to generate an array of signaturecomponents, the hash chain function comprising a series of even-indexhash chains and a series of odd-index has chains, wherein the even-indexhash chains and the odd-index hash chains generate a plurality ofintermediate node values and a one-time public key component between thesecret key components and the signature components; and storing at leastsome of the intermediate node values in the computer-readable memory foruse in one or more subsequent signature operations.

In Example 16, the subject matter of Example 15 can optionally includeinstructions which, when executed by the processor, configure theprocessor to perform operations, comprising computing the message hashof the input message using at least one of a Winterniz One TimeSignature (WOTS) scheme or a WOTS+ scheme that invokes a secure hashalgorithm (SHA) hash function.

In Example 17, the subject matter of any one of Examples 15-16 canoptionally include instructions which, when executed by the processor,configure the processor to perform operations, comprising storing theintermediate node values generated and one time public key componentsgenerated by the even-index hash chains in the computer-readable memory;and using the one-time public key components to define computationsthrough an authentication path through a Merkle tree.

In Example 18, the subject matter of any one of Examples 15-17 canoptionally include instructions which, when executed by the processor,configure the processor to perform operations, comprising applying aL-tree operation to the public key components to compress the public keycomponents into a single leaf node value; and providing the single leafnode value as an input to the Merkle tree.

In Example 19, the subject matter of any one of Examples 15-18 canoptionally include instructions which, when executed by the processor,configure the processor to perform operations, comprising performing afirst hash operations using the single leaf node value and a firstodd-index public key component to generate a first parent node in theMerkle tree; and performing a series of hash operations using subsequentodd-index public key components to determine a root node value of theMerkle tree.

In Example 20, the subject matter of any one of Examples 15-19 canoptionally include instructions which, when executed by the processor,configure the processor to perform operations, comprising comparing theroot node value of the Merkle tree to a multi-signature public key valueassociated with the signing device; and generating an authenticationsignal when the root node value of the Merkle tree matches themulti-signature public key value associated with the signing device.

In Example 21, the subject matter of any one of Examples 15-20 canoptionally include instructions which, when executed by the processor,configure the processor to perform operations, comprising comparing theroot node value of the Merkle tree to a multi-signature public key valueassociated with the signing device; and generating an authenticationfail signal when the root node value of the Merkle tree does not matchthe multi-signature public key value associated with the signing device.

The above Detailed Description includes references to the accompanyingdrawings, which form a part of the Detailed Description. The drawingsshow, by way of illustration, specific embodiments that may bepracticed. These embodiments are also referred to herein as “examples.”Such examples may include elements in addition to those shown ordescribed. However, also contemplated are examples that include theelements shown or described. Moreover, also contemplated are examplesusing any combination or permutation of those elements shown ordescribed (or one or more aspects thereof), either with respect to aparticular example (or one or more aspects thereof), or with respect toother examples (or one or more aspects thereof) shown or describedherein.

Publications, patents, and patent documents referred to in this documentare incorporated by reference herein in their entirety, as thoughindividually incorporated by reference. In the event of inconsistentusages between this document and those documents so incorporated byreference, the usage in the incorporated reference(s) are supplementaryto that of this document; for irreconcilable inconsistencies, the usagein this document controls.

In this document, the terms “a” or “an” are used, as is common in patentdocuments, to include one or more than one, independent of any otherinstances or usages of “at least one” or “one or more.” In addition “aset of” includes one or more elements. In this document, the term “or”is used to refer to a nonexclusive or, such that “A or B” includes “Abut not B,” “B but not A,” and “A and B,” unless otherwise indicated. Inthe appended claims, the terms “including” and “in which” are used asthe plain-English equivalents of the respective terms “comprising” and“wherein.” Also, in the following claims, the terms “including” and“comprising” are open-ended; that is, a system, device, article, orprocess that includes elements in addition to those listed after such aterm in a claim are still deemed to fall within the scope of that claim.Moreover, in the following claims, the terms “first,” “second,” “third,”etc. are used merely as labels, and are not intended to suggest anumerical order for their objects.

The terms “logic instructions” as referred to herein relates toexpressions which may be understood by one or more machines forperforming one or more logical operations. For example, logicinstructions may comprise instructions which are interpretable by aprocessor compiler for executing one or more operations on one or moredata objects. However, this is merely an example of machine-readableinstructions and examples are not limited in this respect.

The terms “computer readable medium” as referred to herein relates tomedia capable of maintaining expressions which are perceivable by one ormore machines. For example, a computer readable medium may comprise oneor more storage devices for storing computer readable instructions ordata. Such storage devices may comprise storage media such as, forexample, optical, magnetic or semiconductor storage media. However, thisis merely an example of a computer readable medium and examples are notlimited in this respect.

The term “logic” as referred to herein relates to structure forperforming one or more logical operations. For example, logic maycomprise circuitry which provides one or more output signals based uponone or more input signals. Such circuitry may comprise a finite statemachine which receives a digital input and provides a digital output, orcircuitry which provides one or more analog output signals in responseto one or more analog input signals. Such circuitry may be provided inan application specific integrated circuit (ASIC) or field programmablegate array (FPGA). Also, logic may comprise machine-readableinstructions stored in a memory in combination with processing circuitryto execute such machine-readable instructions. However, these are merelyexamples of structures which may provide logic and examples are notlimited in this respect.

Some of the methods described herein may be embodied as logicinstructions on a computer-readable medium. When executed on aprocessor, the logic instructions cause a processor to be programmed asa special-purpose machine that implements the described methods. Theprocessor, when configured by the logic instructions to execute themethods described herein, constitutes structure for performing thedescribed methods. Alternatively, the methods described herein may bereduced to logic on, e.g., a field programmable gate array (FPGA), anapplication specific integrated circuit (ASIC) or the like.

In the description and claims, the terms coupled and connected, alongwith their derivatives, may be used. In particular examples, connectedmay be used to indicate that two or more elements are in direct physicalor electrical contact with each other. Coupled may mean that two or moreelements are in direct physical or electrical contact. However, coupledmay also mean that two or more elements may not be in direct contactwith each other, but yet may still cooperate or interact with eachother.

Reference in the specification to “one example” or “some examples” meansthat a particular feature, structure, or characteristic described inconnection with the example is included in at least an implementation.The appearances of the phrase “in one example” in various places in thespecification may or may not be all referring to the same example.

The above description is intended to be illustrative, and notrestrictive. For example, the above-described examples (or one or moreaspects thereof) may be used in combination with others. Otherembodiments may be used, such as by one of ordinary skill in the artupon reviewing the above description. The Abstract is to allow thereader to quickly ascertain the nature of the technical disclosure. Itis submitted with the understanding that it will not be used tointerpret or limit the scope or meaning of the claims. Also, in theabove Detailed Description, various features may be grouped together tostreamline the disclosure. However, the claims may not set forth everyfeature disclosed herein as embodiments may feature a subset of saidfeatures. Further, embodiments may include fewer features than thosedisclosed in a particular example. Thus, the following claims are herebyincorporated into the Detailed Description, with each claim standing onits own as a separate embodiment. The scope of the embodiments disclosedherein is to be determined with reference to the appended claims, alongwith the full scope of equivalents to which such claims are entitled.

Although examples have been described in language specific to structuralfeatures and/or methodological acts, it is to be understood that claimedsubject matter may not be limited to the specific features or actsdescribed. Rather, the specific features and acts are disclosed assample forms of implementing the claimed subject matter.

What is claimed is:
 1. An apparatus, comprising: a computer-readable memory; signature logic to: compute a message hash of an input message using a secure hash algorithm; process the message hash to generate an array of secret key components for the input message; apply a hash chain function to the array of secret key components to generate an array of signature components, the hash chain function comprising a series of even-index hash chains and a series of odd-index has chains, wherein the even-index hash chains and the odd-index hash chains generate a plurality of intermediate node values and a one-time public key component between the secret key components and the signature components; and store at least some of the intermediate node values in the computer-readable memory for use in one or more subsequent signature operations.
 2. The apparatus of claim 1, further comprising logic to: compute the message hash of the input message using at least one of a Winterniz One Time Signature (WOTS) scheme or a WOTS+ scheme that invokes a secure hash algorithm (SHA) hash function.
 3. The apparatus of claim 1, further comprising logic to: store the intermediate node values generated and one time public key components generated by the even-index hash chains in the computer-readable memory; and use the one-time public key components to define computations through an authentication path through a Merkle tree.
 4. The apparatus of claim 2, the signature logic to: apply a L-tree operation to the public key components to compress the public key components into a single leaf node value; and provide the single leaf node value as an input to the Merkle tree.
 5. The apparatus of claim 4, the signature logic to: perform a first hash operations using the single leaf node value and a first odd-index public key component to generate a first parent node in the Merkle tree; and perform a series of hash operations using subsequent odd-index public key components to determine a root node value of the Merkle tree.
 6. The apparatus of claim 5, comprising logic to: compare the root node value of the Merkle tree to a multi-signature public key value associated with the signing device; and generate an authentication success signal when the root node value of the Merkle tree matches the multi-signature public key value associated with the signing device.
 7. The apparatus of claim 5, wherein: compare the root node value of the Merkle tree to a multi-signature public key value associated with the signing device; and generate an authentication fail signal when the root node value of the Merkle tree does not match the multi-signature public key value associated with the signing device.
 8. A computer-implemented method, comprising: computing a message hash of an input message using a secure hash algorithm; processing the message hash to generate an array of secret key components for the input message; applying a hash chain function to the array of secret key components to generate an array of signature components, the hash chain function comprising a series of even-index hash chains and a series of odd-index hash chains, wherein the even-index hash chains and the odd-index hash chains generate a plurality of intermediate node values and a one-time public key component between the secret key components and the signature components; and storing at least some of the intermediate node values in the computer-readable memory for use in one or more subsequent signature operations.
 9. The method of claim 8, further comprising: computing the message hash of the input message using at least one of a Winterniz One Time Signature (WOTS) scheme or a WOTS+ scheme that invokes a secure hash algorithm (SHA) hash function.
 10. The method of claim 9, further comprising: storing the intermediate node values generated and one time public key components generated by the even-index hash chains in the computer-readable memory; and using the one-time public key components to define computations through an authentication path through a Merkle tree.
 11. The method of claim 10, further comprising: applying a L-tree operation to the public key components to compress the public key components into a single leaf node value; and providing the single leaf node value as an input to the Merkle tree.
 12. The method of claim 11, further comprising: performing a first hash operations using the single leaf node value and a first odd-index public key component to generate a first parent node in the Merkle tree; and performing a series of hash operations using subsequent odd-index public key components to determine a root node value of the Merkle tree.
 13. The method of claim 12, further comprising: comparing the root node value of the Merkle tree to a multi-signature public key value associated with the signing device; and generating an authentication signal when the root node value of the Merkle tree matches the multi-signature public key value associated with the signing device.
 14. The method of claim 12, further comprising: comparing the root node value of the Merkle tree to a multi-signature public key value associated with the signing device; and generating an authentication fail signal when the root node value of the Merkle tree does not match the multi-signature public key value associated with the signing device.
 15. A non-transitory computer-readable medium comprising instructions which, when executed by a processor, configure the processor to perform operations, comprising: storing a public key associated with a signing device in a computer-readable medium; computing a message hash of an input message using a secure hash algorithm; processing the message hash to generate an array of secret key components for the input message; applying a hash chain function to the array of secret key components to generate an array of signature components, the hash chain function comprising a series of even-index hash chains and a series of odd-index hash chains, wherein the even-index hash chains and the odd-index hash chains generate a plurality of intermediate node values and a one-time public key component between the secret key components and the signature components; and storing at least some of the intermediate node values in the computer-readable memory for use in one or more subsequent signature operations.
 16. The non-transitory computer-readable medium of claim 15, further comprising instructions which, when executed by the processor, configure the processor to perform operations, comprising: computing the message hash of the input message using at least one of a Winternitz One Time Signature (WOTS) scheme or a WOTS+ scheme that invokes a secure hash algorithm (SHA) hash function.
 17. The non-transitory computer-readable medium of claim 16, further comprising instructions which, when executed by the processor, configure the processor to perform operations, comprising: storing the intermediate node values generated and one time public key components generated by the even-index hash chains in the computer-readable memory; and using the one-time public key components to define computations through an authentication path through a Merkle tree.
 18. The non-transitory computer-readable medium of claim 17, further comprising instructions which, when executed by the processor, configure the processor to perform operations, comprising: applying a L-tree operation to the public key components to compress the public key components into a single leaf node value; and providing the single leaf node value as an input to the Merkle tree.
 19. The non-transitory computer-readable medium of claim 18, further comprising instructions which, when executed by the processor, configure the processor to perform operations, comprising: performing a first hash operations using the single leaf node value and a first odd-index public key component to generate a first parent node in the Merkle tree; and performing a series of hash operations using subsequent odd-index public key components to determine a root node value of the Merkle tree.
 20. The non-transitory computer-readable medium of claim 19, further comprising instructions which, when executed by the processor, configure the processor to perform operations, comprising: comparing the root node value of the Merkle tree to a multi-signature public key value associated with the signing device; and generating an authentication signal when the root node value of the Merkle tree matches the multi-signature public key value associated with the signing device.
 21. The non-transitory computer-readable medium of claim 19, further comprising instructions which, when executed by the processor, configure the processor to perform operations, comprising: comparing the root node value of the Merkle tree to a multi-signature public key value associated with the signing device; and generating an authentication fail signal when the root node value of the Merkle tree does not match the multi-signature public key value associated with the signing device. 